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[EQUATION] The connection [MATH] induces a connection [MATH] -form on [MATH] and a splitting [MATH] into horizontal and vertical spaces; see Section for background. |
{defi} The Kaluza-Klein metric on [MATH] is the [MATH] -invariant metric [MATH] such that the horizontal space [MATH] is isometric to [MATH] , so that [MATH] is orthogonal to [MATH] and is invariant under the natural [MATH] action and so that the orbits of the [MATH] action are unit speed vertical geodesics. |
The data [MATH] determines a horizontal Laplacian [MATH] , a vertical Laplacian [MATH] , and their sum, the Kaluza-Klein Laplacian 1.1 ), |
[EQUATION] As is well-known, sections [MATH] of powers [MATH] of a complex line bundle [MATH] lift to equivariant functions [MATH] |
on the dual line bundle by the formula [MATH] where a point of [MATH] is denoted by [MATH] Under this identification, the horizontal Laplacian is equivalent to the Bochner Laplacians [MATH] on sections of [MATH] Thus, equivariant eigenfunctions of [MATH] of weight [MATH] on [MATH] are lifts of eigensections of [MATH] .... |
and Lemma 6.3 for details. In proving genericity theorems it is easier to work downstairs on [MATH] . But the nodal results pertain to the equivariant eigenfunctions on [MATH] |
{rema} Not all [MATH] invariant metrics on [MATH] are adapted Kaluza-Klein metrics (see Section 6.1 ). Many of the techniques of this paper extend with no major modifications to general [MATH] -invariant metrics on principal [MATH] bundles over manifolds of all dimensions. For simplicity of exposition we restrict to di... |
1.2. Nodal sets We thus have two versions of the eigenfunctions of the Kaluza-Klein [MATH] first as scalar complex valued equivariant eigenfunctions on [MATH] and second as complex eigensections on [MATH] . In each version we have a nodal set, and we use the base nodal set on [MATH] to analyse the nodal set on [MATH] |
We denote the eigensection corresponding to [MATH] as [MATH] in a local holomorphic frame. We mainly consider [MATH] and then we write the section as [MATH] Let |
[EQUATION] Then, [EQUATION] so that with [MATH] [EQUATION] See Section 6.2 for more details. We denote by [MATH] the zero set of the eigensection [MATH] on [MATH] |
[EQUATION] It is easy to see that the zero set [MATH] of [MATH] is the inverse image of [MATH] under the natural projection [MATH] |
[EQUATION] Usually we study the nodal sets of the real and imaginary parts of the lift, not to be confused with the lifts of the real and imaginary parts of the local expression [MATH] of the section (since the frame [MATH] must also be taken into account). In general, it is not obvious whether or not the zero set of [... |
We denote the nodal sets of the real, resp. imaginary parts, of the lift by [EQUATION] The analysis is the same for real and imaginary parts and we generally work with the imaginary part, following the tradition for quadratic differentials. |
Perhaps the most familiar setting for such Kaluza-Klein metrics and lifts of [MATH] -differentials to equivariant sections is that of hyperbolic surfaces of finite area. The reader familiar with Maass forms and operators may want to compare Kaluza-Klein notions with those of [MATH] -theory in Section 9.3 |
1.3. Statement of results Let [MATH] be a Riemannian surface with complex structure [MATH] . Let [MATH] be a Hermitian holomorphic line bundle over [MATH] , and let |
[MATH] be the principal [MATH] bundle associated to [MATH] . Let [MATH] be an [MATH] compatible connection on [MATH] , and let [MATH] be the associated Kaluza-Klein metric on [MATH] |
As mentioned above, weight [MATH] (invariant) eigenfunctions are special because they are real-valued (once they are multiplied by a suitable constant). We therefore separate the case [MATH] from the remainder of the discussion, and state the obvious (but interesting) |
Proposition 1 For [MATH] , invariant eigenfunctions ( [MATH] ) of [MATH] are lifts [MATH] of eigenfunctions [MATH] of [MATH] on the base [MATH] , and the nodal set of |
[MATH] is the inverse image under [MATH] of the nodal set of [MATH] . The number of nodal domains of [MATH] equals the number of nodal domains of [MATH] |
Indeed, their nodal sets are inverse images of nodal sets on the base. Hence the number of nodal domains of ‘invariant’ Kaluza-Klein eigenfunctions is the number for the corresponding eigenfunction on the base. Henceforth we always assume [MATH] |
To prepare for our main result when [MATH] , we first state a result on generic properties of equivariant Kaluza-Klein eigenfunctions. By ‘generic’ properties of Kaluza-Klein metrics, we mean properties of residual sets in a suitable [MATH] space of the data [MATH] , often when only one component is varied and the othe... |
, and the reader is referred there for background. Somewhat surprisingly, generic properties of eigensections of Bochner-Kodaira operators on complex line bundles do not seem to have been studied before. |
The most general genericity results are stated in Theorem in Section . In these results, we define ‘admissible data’ such as [MATH] , and prove the main genericity properties as different components of the data is varied. Since the general result requires some definitions from Sections , we only state the most elementa... |
{theo} Let [MATH] be a Riemann surface, and let [MATH] . We consider Riemannian metrics [MATH] in the conformal class associated to [MATH] . We assume that [MATH] is the Hermitian metric induced by [MATH] and that [MATH] is the Levi-Civita connection. Then, for generic metrics [MATH] in the class of [MATH] on [MATH] |
(1) the spectrum of each Bochner Laplacians [MATH] on [MATH] is simple (i.e. of multiplicity [MATH] ). Thus, the multiplicity of the eigenvalue [MATH] of [MATH] is [MATH] if [MATH] , and [MATH] if [MATH] |
(2) Every eigenfunction is a joint eigenfunction of [MATH] and [MATH] (3) all of the eigensections [MATH] have isolated zeros and zero is a regular value. In particular, [MATH] is a finite set of points; |
(4) If we lift sections to equivariant eigenfunctions [MATH] then [MATH] and [MATH] have zero as a regular value. An important consequence of (1)-(2) of Theorem 1.3 is that, for generic data, all real Kaluza-Klein eigenfunctions are real/imaginary parts of equivariant eigenfunctions. Therefore, the results we prove for... |
We can now state our main result. The first statement repeats (1)-(2) of Theorem 1.3 for the sake of clarity. {theo} Suppose that the data [MATH] of the Kaluza-Klein metric satisfies the generic properties of Theorem 1.3 . Then, |
(1) The eigenspace of [MATH] corresponding to [MATH] is spanned by [MATH] and [MATH] . In particular, any real eigenfunction with the eigenvalue [MATH] is a constant multiple of [MATH] , where [MATH] is the [MATH] action on [MATH] parameterized by [MATH] |
(2) For [MATH] , the nodal sets of [MATH] are connected. (3) For [MATH] , the number of nodal domains of [MATH] is [MATH] Note from Weyl law that [MATH] and that [MATH] . Therefore as an immediate consequence of Theorem 1.3 , we have the following: |
{coro} Let [MATH] be a non-trivial principal [MATH] bundle with a generic Kaluza-Klein metric. For any given orthonormal eigenbasis, almost all (i.e., along a subsequence of density one) eigenfunctions have exactly two nodal domains. |
The density one subsequence is of course the one with [MATH] Theorem 1.3 furnishes the first example of Riemannian manifolds of dimension [MATH] for which the number of nodal domains and connected components of the nodal set have been counted precisely. The results for [MATH] may seem rather surprising, since in dimens... |
and those of Stern on a flat torus . In those cases, the separation-of-variables eigenfunctions have connected nodal sets but the complement of the nodal set has many components, i.e., nodal domains, saturating the Courant bound that the number of nodal domains of the [MATH] th eigenfunction (in order of increasing eig... |
{rema} When [MATH] is trivial, and [MATH] is endowed with the product metric, we have [MATH] where [MATH] is an eigenfunction of [MATH] on the base [MATH] . Hence [MATH] has many nodal domains, and the last statement in Theorem 1.3 fails. Hence, the ‘generic’ set of metrics is not the full set of metrics. See Section 9... |
1.4. Outline of the proof The nodal set of the lift real and imaginary parts of the lift [MATH] of [MATH] is very different over nodal versus non-nodal points of [MATH] Let |
[EQUATION] be the inverse image of the base nodal points. It is a union of fibers and is a finite union of fibers if and only if [MATH] has a finite number of zeros. We refer to [MATH] as the ‘singular fibers’ or singular set. We denote by [MATH] the punctured Riemann surface in which the zero set of [MATH] is deleted.... |
Proposition 2 For [MATH] , the maps [EQUATION] is an [MATH] -fold covering space. It follows that the topology of the nodal set is entirely determined by the combinatorics of gluing the sheets along the singular fibers. In fact, the gluing is rather simple and easily yields the following |
{theo} For all [MATH] , the nodal set [MATH] is connected. To count nodal domains, we need to make the assumption that there are just a finite number of zeros of [MATH] and that at least one of them is regular. |
When the zero set is transverse to the zero section, then the sum of the indices of the zeros is the first Chern class of [MATH] , and in particular is non-empty when the genus of [MATH] is [MATH] , i.e., when [MATH] is not a torus. |
For metrics satisfying Theorem 1.3 we prove Theorem 1.3 by using Proposition together with some geometric observations on how the sheets fit together at the singular fibers. This is done using a Bers type local analysis of the eigensections (Section ) and some geometric/combinatorial arguments in Section |
To put the nodal results into context, it is proved in varying degrees of generality in that in dimension [MATH] , the number of nodal domains of an orthonormal basis [MATH] of Laplace eigenfunctions on certain surfaces with ergodic geodesic flow tends to infinity with the eigenvalue along almost the entire sequence of... |
{rema} Note that the geodesic flow on [MATH] with a Kaluza–Klein metric never is ergodic. To see this, observe that the hypersurface |
[EQUATION] is invariant under the geodesic flow, and it divides [MATH] into two subsets of positive measure: [EQUATION] 1.4.1. Surfaces of constant curvature |
Metrics [MATH] of constant curvature with their associated Hermitian metrics and connections on [MATH] are not generic. But they are of special interest, so we comment on what we are able to prove about them. Note that the standard metric on [MATH] is Kaluza-Klein, as is the standard metric on [MATH] or [MATH] . The st... |
Perhaps surprisingly, the results of Theorem 1.3 are valid for some orthonormal bases of eigenfunctions on flat [MATH] -tori. {theo} |
On the flat [MATH] torus [MATH] , one can find an orthonormal eigenbasis for which all nonconstant eigenfunctions have two nodal domains. |
Next we turn to hyperbolic metrics on a surface [MATH] of genus [MATH] Then the total space [MATH] and the equivariant eigenfunctions of the Kaluza-Klein Laplacian [MATH] are the same as joint eigenfunctions of the generator [MATH] of [MATH] and of the Casimir operator |
[MATH] When the weight [MATH] is fixed, one may separate variables and obtain a Maass Laplacian [MATH] on smooth sections of a complex line bundle [MATH] , namely the bundle of [MATH] -differentials of type [MATH] , and are the usual weight [MATH] automorphic Maass eigendifferentials [MATH] |
[EQUATION] of the Maass Laplacians [EQUATION] Unfortunately, we are not able to verify that (any) Maass eigendifferentials have [MATH] as a regular value, i.e. the generic conditions needed for Theorem 1.3 . Indeed, we do not know how to prove that (any) eigenfunctions [MATH] of the hyperbolic Laplacian have a discrete... |
Regarding spherical harmonics on [MATH] we will be brief, because we study random equivariant spherical harmonics in detail in a forthcoming article |
. The round metric is a Kaluza-Klein metric, but of course not a generic one. In Section 9.2 we show that the joint eigenfunctions [MATH] of [MATH] and of two commuting [MATH] actions have very different nodal sets from the ones in Theorem 1.3 . On the other hand, in |
we show that ‘random’ linear combinations of such [MATH] with [MATH] fixed do satisfy the results of Theorem 1.3 and the nodal sets of their real, resp. imaginary parts, have just one nodal component. We also find their expected Euler characteristic. These results were motivated by the numerical discovery of Barnett et... |
We are fixing the weight [MATH] and therefore do not work with general random spherical harmonics. But in the [MATH] dimensional subspaces where |
[MATH] is fixed with [MATH] , the nodal sets of the real and imaginary parts are connected and divide [MATH] into just two components. For further discussion we refer to |
2. Geometric background In this section we discuss the geometric data that goes into the construction of Kaluza-Klein metrics, which are defined in Definition |
1.1 . They are also the data needed to define Bochner Laplacians [MATH] and Kaluza-Klein Laplacians [MATH] We plan to vary the data and study perturbation theory of eigenvalues and eigensections in Section |
2.1. Riemannian metrics on [MATH] and Hermitian metrics on [MATH] Let [MATH] denote a Riemann surface with complex structure [MATH] and Riemannian metric [MATH] . We write [MATH] and [MATH] , where [MATH] is the dual metric. The complex structure gives a decomposition of [MATH] into [MATH] resp. [MATH] parts. We denote... |
[EQUATION] where the Kähler form [MATH] is the [MATH] form defined by [MATH] . Locally there exists a Kähler potential [MATH] defined up to a constant by [MATH] Here [MATH] . Then, [MATH] , where [MATH] |
2.1.1. The space of isometry classes of metrics on surfaces It is well-known that the space of Riemannian metric tensors on a manifold [MATH] splits as a product [MATH] of volume forms times metrics with a fixed volume form. Thus, one may separately consider metrics with a fixed volume form and conformal classes of met... |
Choice of a complex structure [MATH] on [MATH] is equivalent to choice of a conformal class [MATH] of metrics. The moduli space of conformal classes is the same as the [MATH] -dimensional moduli space [MATH] of complex structures on [MATH] . In each conformal class, we may pick a background metric [MATH] and represent ... |
[EQUATION] If we fix a complex structure [MATH] , then the Riemannian metrics in the corresponding conformal class are Kähler metrics, and may be parameterized by their Kähler potentials [MATH] , where the area form [MATH] of the Kähler metric is related to that of the reference metric by |
[EQUATION] We let [EQUATION] which may be identified with an open set in [MATH] . The Liouville field and Kähler potential are related by |
[EQUATION] where [MATH] is the Laplacian of [MATH] . The only difference in the two parameterizations of conformal metrics is that the area of metrics in [MATH] is fixed while it may vary in [MATH] . Thus [MATH] |
In the case of Riemann surfaces, the area form is the symplectic form associated to a Kähler metric. Given a complex structure [MATH] , the Kähler metric [MATH] can be recovered from its area form [MATH] |
by the formula [MATH] . Hence isometry classes of Kähler metrics with a fixed area form are parameterized by [MATH] 2.2. Complex line bundles [MATH] , connections and curvature |
If we fix a complex structure [MATH] and holomorphic line bundle [MATH] then a Hermitian metric [MATH] on [MATH] is determined by the length of a local holomorphic frame [MATH] (i.e., a local holomorphic nonvanishing section) of [MATH] over an open set [MATH] by [MATH] , where |
[MATH] denotes the [MATH] -norm of [MATH] 2.2.1. Connections In the real setting, a connection on a vector bundle [MATH] defines a covariant derivative |
[EQUATION] In our complex setting, we assume [MATH] is a holomorphic Hermitian line bundle, i.e., we equip [MATH] with a complex structure [MATH] , a connection [MATH] , and a Hermitian metric [MATH] . In a local frame [MATH] it is defined by [MATH] . We consider several types of compatibility conditions between this d... |
An [MATH] -connection [MATH] is one compatible with [MATH] . In a unitary frame, the connection [MATH] -form is [MATH] valued and is denoted by [MATH] . We denote the space of [MATH] -compatible connections by [MATH] |
Or a [MATH] -compatible connection. In a holomorphic frame [MATH] the connection [MATH] -form [MATH] is of type [MATH] . We denote the space of [MATH] -compatible connections by [MATH] |
The unique Chern connection [MATH] which is compatible with both [MATH] This data induces: The Hermitian metric [MATH] induces the principal bundle of [MATH] -unitary frames [MATH] |
An [MATH] -compatible connection [MATH] induces a real [MATH] -form [MATH] on [MATH] Connections [MATH] determine complex-valued [MATH] -forms on [MATH] |
The connection 1-form in the frame [MATH] is given by [EQUATION] We denote the [MATH] resp. [MATH] parts of [MATH] by [MATH] resp. [MATH] |
Suppose that [MATH] . Then if [MATH] with [MATH] a local holomorphic frame, [EQUATION] The holomorphic line bundle [MATH] also has a natural Cauchy-Riemann operator, |
[EQUATION] In a local holomorphic frame [MATH] , we write a smooth section [MATH] and then [EQUATION] It is well-defined since if [MATH] is another holomorphic frame and |
[MATH] , then [MATH] and [MATH] The Chern connection [MATH] associated to the Hermitian metric [MATH] is the unique metric connection |
[EQUATION] whose connection [MATH] -form in a holomorphic frame [MATH] has type [MATH] The connection 1-form is given by [MATH] with [MATH] |
The metric [MATH] is a Hermitian metric on [MATH] . Any Hermitian metric [MATH] on a line bundle [MATH] induces metrics [MATH] on the tensor powers [MATH] in the local frame [MATH] The Hermitian metric and complex structure determine a Chern connection |
[MATH] whose curvature 2-form [MATH] is given locally by [EQUATION] and we say that [MATH] is positive if the (real) 2-form [MATH] is positive. |
2.3. Curvature form Given a connection [MATH] on [MATH] and a vector field [MATH] on [MATH] the covariant derivative of a section [MATH] is defined by [MATH] . The curvature is the 2-form [MATH] |
defined by [MATH] If [MATH] is a local frame and [MATH] then [MATH] 2.4. Examples Let [MATH] be the unit co-frame bundle [MATH] , consisting of orthonormal frames of [MATH] Then [MATH] is the bundle of real [MATH] -differentials, i.e., homogeneous polynomials of degree [MATH] |
in [MATH] or [MATH] When [MATH] is given a complex structure, we may decompose [MATH] into co-vectors [MATH] of type [MATH] and [MATH] of type [MATH] . The holomorphic tangent bundle is usually denoted by [MATH] and is called the canonical bundle. Its tensor powers [MATH] are bundles of differentials of type [MATH] wit... |
When the genus is [MATH] , i.e., [MATH] [MATH] is a negative line bundle and has no holomorphic sections. The associated circle bundle of frames is [MATH] |
When the genus is [MATH] , then [MATH] and [MATH] are trivial and [MATH] When the genus is [MATH] we may twist [MATH] by a flat line bundle. This is not particularly relevant for this article except that we usually ignore this additional degree of freedom. There is an ample line bundle [MATH] whose holomorphic sections... |
When the genus is [MATH] then [MATH] where [MATH] . The associated [MATH] bundle is [MATH] [MATH] is ample and for [MATH] large there are many holomorphic sections of [MATH] . This is only significant in this article when we discuss splitting eigenspaces. |
2.5. Canonical bundle and [MATH] -differentials Let [MATH] denote the canonical bundle [MATH] of [MATH] forms [MATH] . Up to twisting by a flat line bundle, it is the unique ample line bundle on [MATH] . Hence there exists a Hermitian metric [MATH] on [MATH] with curvature form [MATH] . This should be distinguished fro... |
[EQUATION] In terms of the Hermitian metric [MATH] on [MATH] [MATH] . Also, [EQUATION] We now regard [MATH] as a Kähler metric. The co-metric defines metric coefficients on [MATH] by extending [MATH] by complex linearity and induces the Hermitian metric, |
[EQUATION] on [MATH] The curvature [MATH] form is therefore [MATH] The associated [MATH] form [MATH] is positive if the genus is [MATH] and if [MATH] is a metric of negative curvature [MATH] . It is negative if the genus is [MATH] and the metric [MATH] is of positive curvature. When the genus is [MATH] there do not exi... |
When [MATH] we write the area form as [MATH] or as [MATH] . The metric [MATH] induces metrics [MATH] on [MATH] and on powers such as [MATH] . On [MATH] , the Hermitian metric induced by |
[MATH] is [MATH] so [MATH] The Chern connection on [MATH] is the same as the Riemannian connection. Consider the complex line vector bundles [MATH] and [MATH] . They are isomorphic under the map |
[EQUATION] {lemm} Let [MATH] be a Kähler manifold. Under the isomorphism [MATH] , the Chern connection [MATH] on the holomorphic tangent bundle [MATH] |
is the Levi-Civita connection [MATH] 2.6. Orthonormal frame bundles and [MATH] -differentials If [MATH] is a Riemannian surface, and [MATH] is the [MATH] -bundle of orthonormal frames of [MATH] , then |
[MATH] determines a Riemannian connection on [MATH] . Similarly, if we fix a complex structure and define the principal [MATH] bundle [MATH] associated to [MATH] , then [MATH] determines a Hermitian metric on [MATH] . We first discuss the real geometry and then the complex geometry. |
The metric [MATH] on [MATH] induces a co-metric [MATH] on [MATH] , usually denoted by raised indices. It then induces metrics [MATH] on powers [MATH] |
There always exists a basis of basic or horizontal [MATH] -forms at a frame [MATH] such that [EQUATION] The Riemannian connection [MATH] -form [MATH] is defined by the equations, |
[EQUATION] where [MATH] is the scalar curvature. Dually, there exist vector fields [MATH] so that [EQUATION] Then define [EQUATION] |
Then, [EQUATION] In the frame [MATH] the volume form is [MATH] The vector fields [MATH] are also of unit length since they are defined to be horizontal lifts of a unit frame. |
2.7. Hilbert spaces of sections Let [MATH] be a Hermitian holomorphic line bundle. We thus have a pair of metrics, [MATH] resp. [MATH] (with Kähler form [MATH] ) on [MATH] resp. [MATH] |
To each pair [MATH] of metrics we associate Hilbert space inner products [MATH] on sections [MATH] of the form [EQUATION] where [MATH] is the pointwise Hermitian norm-squared of the section [MATH] in the metric [MATH] . In a local holomorphic frame [MATH] , we write |
[EQUATION] In local coordinates [MATH] and the local frame [MATH] of [MATH] , we may write [MATH] and then [EQUATION] Henceforth we write |
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